Systematic MTC catalog

Terry Gannon, Gerald Höhn, Hiroshi Yamauchi, ...

Table of content

• prime MTCs by rank
• prime MTCs by discriminant
• finite quadratic space MTCs
• finite group orbifolds
• MTCs with interesting genera
• Kac-Moody MTCs

prime MTCs up to conjugation by rank (with same S-matrices enclosed in brackets)

• 1 simple objects: trivial
• 2 simple objects: qs2, lee_yang
• 3 simple objects: qs3, (ising1, ising2, ising3, ising4), 3fieldsx
• 4 simple objects: (qs4, qn4), (qu2, qv2), 4fieldsx
• 5 simple objects: (qs5, qn5), (kmA1_4, min3fac5), 5fieldsx, kmF4_2
• 6 simple objects: kmA2_2, (kmE7_2, N1min1), kmB2_2, kmG2_3
• 7 simple objects: qs7, (kmC6_1, kmA1_6), kmB3_2
• 8 simple objects: (qs8, qn8), kmB4_2
• 9 simple objects: qs9, kmB5_2, kmC8_1, (kmF4_3, kmG2_4),
• 10 simple objects: kmA2_3, kmB6_2, (kmB2_3, kmC3_2), e6double, kmE8_4
• 11 simple objects: qs11, kmD4_2, kmB7_2, (kmA1_10, kmC10_1)
• 12 simple objects: kmB8_2, haagerup12, kmG2_5

prime MTCs up to conjugation by discriminant (with same S-matrices enclosed in brackets)

• disc.=1: trivial
• disc.=2: qs2
• disc.=3: qs3
• disc.=(5 + Sqrt[5])/2: lee_yang
• disc.=4: (qs4, qn4), (qu2, qv2), (ising1, ising2, ising3, ising4)
• disc.=5: (qs5, qn5)
• disc.=7: qs7
• disc.=8: (qs8, qn8)
• disc.=9: qs9
• disc.=(7*Csc[Pi/7]^2)/4: 3fieldsx
• disc.=11: qs11
• disc.=12: (kmA1_4, min3fac5)
• disc.=Root[-7 + 21*#1 - 14*#1^2 + #1^3 & , 3, 0] : 5fieldsx
• disc.=13: qs13, qn13
• disc.=16: qs16, qs16-, qu4, qv4
• disc.=17: qs17, qn17
• disc.=18.5917938865 : kmA1_4
• disc.=19: qs19
• disc.=19.2344223834 : kmG2_2
• disc.=20: kmB2_2

prime finite quadratic space MTCs

• prime MTCs: trivial, qs2, qs2-, qs3, qs3-, qs4, qs4-, qn4, qn4-, qu2, qv2, qs5, qn5, qs7, qs7-, qs8, qs8-, qn8, qn8-, qs9, qs9-, qs11, qs11-, qs13, qn13, qs16, qs16-, qn16, qn16-, qu4, qv4, qs17, qn17, qs19, qs19-, qs23, qs23-, qs25, qn25, qs27, qs27-, qs29, qn29, qs31, qs31-, qs32, qs32-, qn32, qs32-, qs37, qn37, qs41, qn41, qs43, qs43-, qs47, qs47-, qs49, qs49-, qs64, qs64-, qn64, qn64-, qu8, qv8

(Untwisted) finite group orbifolds (up to 63 simple objects)

• abelian groups: Z1, Z2, Z3, Z4, Z2 x Z2, Z5, Z6, Z7,
• symmetric/alternating groups: S3, A4, S4, A5, S5, A6,
• (semi)dihedral/quaternion groups: D5, (D4 (?), Q4 (?)), D7, D9 (?), (D8 (?), Q8 (?), SD8 (?)),
• Frobenius groups: Z5.Z4, Z7.Z3, Z3^2.Z2 (?), Z3^2.Z4, Z3^2.Q4, Z7.Z6, Z11.Z5,
• further named groups: Z2 x S3 (?), DC3 (?), PSL(2,F7), SL(2,F3),
• rest unnamed groups: finiteGr-168-54-1, finiteGr-24-56-1, finiteGr-48-56-1, finiteGr-48-56-2, finiteGr-52-58-1, finiteGr-56-58-1, finiteGr-72-60-1,

MTCs with interesting genera

• 1 simple object: trivial
• 3 simple objects: ising1-
• 4 simple objects: qs2-.lee_yang
• 6 simple objects: lee_yang-.ising4, qs3-.lee_yang, lee_yang.3fieldsx-
• 8 simple objects: lee_yang-.qu2
• 9 simple objects: qs3.3fieldsx-
• 10 simple objects: lee_yang.min3fac5
• 16 simple objects: lee_yang-^2.qs4-
• 1024 simple objects: qu2^5

Kac-Moody MTCs

Type A

• Level 1: A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, A18, A19, A20, A21, A22, A23, A24, ...
• Level 2: A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, A18, A19, A20, A21, A22, A23, A24, ...
• Level 3: A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, ...
• Level 4: A1, A2, A3, A4, A5, A6, A7, A8, A9, ...
• Level 5: A1, A2, A3, A4, A5, A6, A7, A8, A9, ...
• Level 6: A1, A2, A3, A4, ...
• Level 7: A1, A2, A3, A4, ...
• Level 8: A1, A2, A3, A4, ...
• Level 9: A1, A2, A3, A4, ...
• Level 10: A1, A2, A3, A4, ...

Type B

• Level 1: B2, B3, B4, B5, B6, B7, B8, B9, B10, B11, B12, B13, B14, B15, B16, B17, B18, B19, B20, B21, B22, B23, B24, ...
• Level 2: B2, B3, B4, B5, B6, B7, B8, B9, B10, B11, B12, B13, B14, B15, B16, B17, B18, B19, B20, B21, B22, B23, B24, ...
• Level 3: B2, B3, B4, B5, B6, B7, B8, B9, B10, B11, B12, B13, B14, ...
• Level 4: B2, B3, B4, B5, B6, B7, B8, B9, ...
• Level 5: B2, B3, B4, B5, B6, B7, B8, B9, ...
• Level 6: B2, B3, B4, ...
• Level 7: B2, B3, B4, ...
• Level 8: B2, B3, B4, ...
• Level 9: B2, B3, B4, ...
• Level 10: B2, B3, B4, ...

Type C

• Level 1: C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, ...
• Level 2: C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19, C20, C21, C22, C23, C24, ...
• Level 3: C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, ...
• Level 4: C3, C4, C5, C6, C7, C8, C9, ...
• Level 5: C3, C4, C5, C6, C7, C8, C9, ...
• Level 6: C3, C4, ...
• Level 7: C3, C4, ...
• Level 8: C3, C4, ...
• Level 10: C3, C4, ...

Type D

• Level 1: D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, D15, D16, D17, D18, D19, D20, D21, D22, D23, D24, ...
• Level 2: D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, D15, D16, D17, D18, D19, D20, D21, D22, D23, D24, ...
• Level 3: D4, D5, D6, D7, D8, D9, D10, D11, D12, D13, D14, ...
• Level 4: D4, D5, D6, D7, D8, D9, ...
• Level 5: D4, D5, D6, D7, D8, D9, ...
• Level 6: D4, ...
• Level 7: D4, ...
• Level 8: D4, ...
• Level 9: D4, ...
• Level 10: D4, ...

Exceptional types

• E6 at level 1, 2, 3, 4, 5, ...
• E7 at level 1, 2, 3, 4, 5, ...
• E8 at level 1, 2, 3, 4, 5, ...
• F4 at level 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
• G2 at level 1, 2, 3, 4 5, 6, 7, 8 9, 10, ...